
by Cauchy integration formula. The above results can also be deduced from Theorem 5.1.4 of [1]}.
Notice that
\(\widehat{\mathcal {P} }_r (x_n \otimes x_n) \widehat{\mathcal {P} }_s &=(\hat{\theta }_r\otimes \hat{\theta }_r) (x_n \otimes x_n) (\hat{\theta }_s\otimes \hat{\theta }_s) \\&= \langle \hat{\theta }_r, x_n\rangle \cdot \langle \hat{\theta }_s, x_n \rangle \cdot (\hat{\theta }_s\otimes \hat{\theta }_r)= \hat{\sigma }_r \hat{\sigma }_s \hat{\xi }_{ns}\hat{\xi }_{nr} (\hat{\theta }_s\otimes \hat{\theta }_r),\) 
